So we get Rome under Julius Caesar, not an early Kingdom of Rome or a decaying corrupt empire. But the choice of representation isn't quite as simple. There are certain... delicate topics that are avoided. Russia is ruled by Empress Catherine, and its gameplay bonuses are rooted in that time. Surely Russia was more powerful and iconic under Stalin? But that would be awkward. Of course, there's no Hitler or Mussolini either. Germany gets Bismarck, which makes sense, but they couldn't resist putting in WW2-era German tanks. But with sufficient distance, atrocities pale, as evidenced by the inclusion of Genghis Khan.
Beyond the decisions clearly rooted in wanting certain iconic things, and not wanting to cause offence, there's still a lot of places things could have gone differently - different eras that are arguably just as good, and maybe skew towards the more recent. The underlying bias I'm poking at here is that people tend to think of foreign countries as in the past. A particular image of the place gets stuck in people's minds and contaminates their thinking. So yes, they do have Internet access in Namibia, antibiotics in Nepal and roads in France. ("Of course they have roads in France!" you say, but a friend of mine genuinely got asked whether they did about 15 years ago.)
So here's some variant versions of existing civilizations. Most of them are closer to the present day than the originals, and as a result, many of them are probably more controversial. But I'm pretty certain Genghis Khan was... controversial too, in his time.
- American: 20th century USA with its vast cultural influence and military interventionism springs to mind. Some kind of bonus for spreading culture in the late-game? GIs as units?
- Babylonian: Let's pick Saddam Hussein-era Iraq! That won't be controversial at all! They can have a bonus to WMDs, oh, wait.
- Brazilian: The Empire of Brazil lasted for all of 67 years. Of course, subsequent governments lasted even less long, but still... Lula as a high-profile alternative?
- Byzantine: This one I find a bit mysterious. Why not Ataturk, such a towering figure in recent history. Too recent and controversial? Too in love with elephant-based special mechanics?
- Celtic: This civ is a bit of an atrocious mish-mash of different peoples. Are they scots, or picts, or gauls, or what? How about Enlightenment-era Scotland as a science-oriented civ instead?
- English: Well, the obvious alternate choice would be Victorian England, which is really more iconic and more powerful - it's also what everyone else always picks, so Elizabeth is already an exercise in being a bit less obvious.
- French: Going back in time for this one, how about Charlemagne?
- Korean: Modern South Korea would be a pretty obvious pick, though that might cause issues re: its evil twin?
- Persian: Iran under Khomeini is the obvious and controversial alternative.
- Roman: Of course you have to have the Romans, yeah. An alternative would be actual Italy, with Garibaldi as the leader?